We prove that if a pure simplicial complex of dimension d with n facets has the least possible number of (d − 1)-dimensional faces among all complexes with n faces of dimension d, then it is vertex decomposable. This answers a question of J. Herzog and T. Hibi. In fact, we prove a generalization of their theorem using combinatorial methods.

We reformulate Hecke’s open problem of 1923, regarding the Fourier-analytic proof of higher reciprocity laws, as a theorem about morphisms involving stratified topological spaces. We achieve this by placing Kubota’s formulations of n-Hilbert reciprocity in a new topological context, suited to the introduction of derived categories of sheaf complexes. Subsequently, we begin to investigate condit...

We propose an entropy function for simplicial complices. Its value gives the expected cost of the optimal encoding of sequences of vertices of the complex, when any two vertices belonging to the same simplex are indistinguishable. We show that the proposed entropy function can be computed efficiently. By computing the entropy of several complices consisting of hundreds of simplices, we show tha...

Let Γ be a simplicial complex with n vertices, and let ∆(Γ) be either its exterior algebraic shifted complex or its symmetric algebraic shifted complex. If Γ is a simplicial sphere, then it is known that (a) ∆(Γ) is pure and (b) h-vector of Γ is symmetric. Kalai and Sarkaria conjectured that if Γ is a simplicial sphere then its algebraic shifting also satisfies (c) ∆(Γ) ⊂ ∆(C(n, d)), where C(n,...

Historical Background Raster (field) or vector (object) are the two dominant conceptualizations of space. Applications focusing on object with 2 or 3 dimensional geometry structure the storage of geometry as points, lines, surface, and volumes and the relations between them; a classical survey paper discussed the possible approaches mostly from the perspective of Computer Aided Design (CAD) whe...

We formulate and prove inverse mixing lemmas in the settings of simplicial complexes and k-uniform hypergraphs. In the hypergraph setting, we extend results of Bilu and Linial for graphs. In the simplicial complex setting, our results answer a question of Parzanchevski et al.

Given a random 3-uniform hypergraph H = H(n, p) on n vertices where each triple independently appears with probability p, consider the following graph process. We start with the star G0 on the same vertex set, containing all the edges incident to some vertex v0, and repeatedly add an edge xy if there is a vertex z such that xz and zy are already in the graph and xzy ∈ H. We say that the process...

We compare two models for ∞-operads: the complete Segal operads of Barwick and the complete dendroidal Segal spaces of Cisinski and Moerdijk. Combining this with comparison results already in the literature, this implies that all known models for ∞-operads are equivalent — for instance, it follows that the homotopy theory of Lurie’s ∞-operads is equivalent to that of dendroidal sets and that of...